Exercise 9, § 4

Let $f$ be a finite real function, lower semi-continuous on an interval $I \subset \mathbf{R}$. For $f$ to be convex on $I$ it suffices that, for every pair of points $a, b$ of $I$ such that $a < b$ there exists one point $z$ such that $a < z < b$, and that $M_z$ be below the segment $M_aM_b$ (argue by contradiction, noting that the set of points $x$ such that $M_x$ lies strictly above $M_aM_b$ is open).